How to be able to handle both cases of element exists and does not exist?

I would like to have the ability to handle the situation of both cases: the element exists and does not exist.

All my trials ended with an exception when it didn't find the item and therefore I couldn't do the stuff I wanted in this case or errors that using then.catch or then.fail are not valid.

For example, my last trial is:

cy.get('.ant-breadcrumb-link').then(($element) => {
    if ($element.is(':exists')) {
        // Element exists, perform actions
        cy.log('Element exists');
        // ... your actions when the element exists ...
    } else {
        // Element does not exist, perform other actions
        cy.log('Element does not exist');
        // ... your actions when the element does not exist ...
    }
});

So when the cy.get('.ant-breadcrumb-link') fails I don't get into the then() section.

How can I achieve my goal?

You can always leverage jQuery when you face issues with the checking the existence of the element. I would get the document object using cy.document and use Cypress.$(jquery equivalent in cypress) for the same. A sample would look like:

cy.document().then((document) => {
  const $element = Cypress.$('.ant-breadcrumb-link', document);
  
  if ($element.length > 0) {
    // ... your actions when the element exists ...
  } else {
    // ... your actions when the element does not exist ...
  }
});

SIDENOTE:

cy.get() will keep on searching for the element and throws an exception if the element is not found.